Lz2520802095

C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue5, September- October 2012, pp.2080-2095
Thermophoresis Effect On Unsteady Free Convection Heat And
Mass Transfer In A Walters-B Fluid Past A Semi Infinite Plate
C. Sudhakar*, N. Bhaskar Reddy*, #B. Vasu**, V. Ramachandra Prasad**
*Department of Mathematics, Sri Venkateswara University, Tirupati, A.P, India
** Department of Mathematics, Madanapalle Institute of Technology and Sciences, Madanapalle-517325, India.

ABSTRACT
The effect of thermophoresis particle gas flow, the thermophoretic deposition plays an
deposition on unsteady free convective, heat and important role in a variety of applications such as
mass transfer in a viscoelastic fluid along a semi- the production of ceramic powders in high
infinite vertical plate is investigated. The temperature aerosol flow reactors, the production of
Walters-B liquid model is employed to simulate optical fiber performs by the modified chemical
medical creams and other rheological liquids vapor deposition (MCVD) process and in a polymer
encountered in biotechnology and chemical separation. Thermophoresis is considered to be
engineering. The dimensionless unsteady, important for particles of 10 m in radius and
coupled and non-linear partial differential temperature gradient of the order of 5 K/mm.
conservation equations for the boundary layer Walker et al. [1] calculated the deposition efficiency
regime are solved by an efficient, accurate and of small particles due to thermophoresis in a laminar
unconditionally stable finite difference scheme of tube flow. The effect of wall suction and
the Crank-Nicolson type. The behavior of thermophoresis on aerosol-particle deposition from
velocity, temperature and concentration within a laminar boundary layer on a flat plate was studied
the boundary layer has been studied for by Mills et al. [2]. Ye et al. [3] analyzed the
variations in the Prandtl number (Pr), thermophoretic effect of particle deposition on a free
viscoelasticity parameter (), Schmidt number standing semiconductor wafer in a clean room.
(Sc), buoyancy ration parameter (N) and Thakurta et al. [4] computed numerically the
thermophoretic parameter (  ). The local skin- deposition rate of small particles on the wall of a
friction, Nusselt number and Sherwood number turbulent channel flow using the direct numerical
are also presented and analyzed graphically. It is simulation (DNS). Clusters transport and deposition
observed that, an increase in the thermophoretic processes under the effects of thermophoresis were
parameter (  ) decelerates the velocity as well as investigated numerically in terms of thermal plasma
concentration andaccelerates temperature. In deposition processes by Han and Yoshida [5]. In
addition, the effect of the thermophoresis is also their analysis, they found that the thickness of the
discussed for the case of Newtonian fluid. concentration boundary layer was significantly
suppressed by the thermophoretic force and it was
Key words: Finite difference method,semi-infinite concluded that the effect of thermophoresis plays a
vertical plate, Thermophoresis effect, Walters-B more dominant role than that of diffusion. Recently,
fluid, unsteady flow. Alam et al. [6] investigated numerically the effect of
thermophoresis on surface deposition flux on
hydromagnetic free convective heat mass transfer
1. INTRODUCTION
flow along a semi- infinite permeable inclined flat
Prediction of particle transport in non-
isothermal gas flow is important in studying the plate considering heat generation. Their results show
that thermophoresis increases surface mass flux
erosion process in combustors and heat exchangers,
significantly. Recently, Postalnicu [7] has analyzed
the particle behavior in dust collectors and the
fabrications of optical waveguide and the effect of thermophoresis particle deposition in
semiconductor device and so on. Environmental free convection boundary layer from a horizontal
flat plate embedded in porous medium.
regulations on small particles have also become
The study of heat and mass transfer in non-
more stringent due to concerns about atmospheric
Newtonian fluids is of great interest in many
pollution.
operations in the chemical and process engineering
When a temperature gradient is established
industries including coaxial mixers, blood
in gas, small particles suspended in the gas migrate
oxygenators [8], milk processing [9], steady-state
in the direction of decreasing temperature. The
tubular reactors and capillary column inverse gas
phenomenon, called thermophoresis, occurs because
gas molecules colliding on one side of a particle chromatography devices mixing mechanism bubble-
have different average velocities from those on the drop formation processes [10] dissolution processes
other side due to the temperature gradient. Hence and cloud transport phenomena. Many liquids
when a cold wall is placed in the hot particle-laden possess complex shear-stress relationships which

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deviate significantly from the Newtonian (Navier- deformable stretching surface. They found that a
Stokes) model. External thermal convection flows in boundary layer is generated formed when the
such fluids have been studied extensively using inviscid free-stream velocity exceeds the stretching
mathematical and numerical models and often velocity of the surface and the flow is accelerated
employ boundary-layer theory. Many geometrical with increasing magnetic field. This study also
configurations have been addressed including flat identified the presence of an inverted boundary layer
plates, channels, cones, spheres, wedges, inclined when the surface stretching velocity exceeds the
planes and wavy surfaces. Non-Newtonian heat velocity of the free stream and showed that for this
transfer studies have included power-law fluid scenario the flow is decelerated with increasing
models [11-13] i.e. shear-thinning and shear magnetic field. Rajagopal et al [32] obtained exact
thickening fluids, simple viscoelastic fluids [14, 15], solutions for the combined nonsimilar
Criminale-Ericksen-Fibley viscoelastic fluids [16], hydromagnetic flow, heat, and mass transfer
Johnson-Segalman rheological fluids [17], Bingham phenomena in a conducting viscoelastic Walters-B
yield stress fluids [18], second grade (Reiner-Rivlin) fluid percolating a porous regime adjacent to a
viscoselastic fluids [19] third grade viscoelastic stretching sheet with heat generation, viscous
fluids [20], micropolar fluids [21] and bi-viscosity dissipation and wall mass flux effects, using
rheological fluids [22]. Viscoelastic properties can confluent hypergeometricfunctions for different
enhance or depress heat transfer rates, depending thermal boundary conditions at the wall.
upon the kinematic characteristics of the flow field Steady free convection heat and mass
under consideration and the direction of heat transfer flow of an incompressible viscous fluid past
transfer. The Walters-B viscoelastic model [23] was an infinite or semi-infinite vertical plate is studied
developed to simulate viscous fluids possessing since long because of its technological importance.
short memory elastic effects and can simulate Pohlhausen [33], Somers [34] and Mathers et al.
accurately many complex polymeric, [35] were the first to study it for a flow past a semi-
biotechnological and tribological fluids. The infinite vertical plate by different methods. But the
Walters-B model has therefore been studied first systematic study of mass transfer effects on free
extensively in many flow problems. Soundalegkar convection flow past a semi-infinite vertical plate
and Puri[24] presented one of the first mathematical was presented by Gebhart and pera [36] who
investigations for such a fluid considering the presented a similarity solution to this problem and
oscillatory two-dimensional viscoelastic flow along introduced a parameter N which is a measure of
an infinite porous wall, showing that an increase in relative importance of chemical and thermal
the Walters elasticity parameter and the frequency diffusion causing a density difference that drives the
parameter reduces the phase of the skin-friction. flow. Soundalgekar and Ganesan [37] studied
Roy and Chaudhury [25] investigated heat transfer transient free convective flow past a semi-infinite
in Walters-B viscoelastic flow along a plane wall vertical flat plate with mass transfer by using
with periodic suction using a perturbation method Crank–Nicolson finite difference method. In their
including viscous dissipation effects. Raptis and analysis they observed that, an increase in N leads to
Takhar [26] studied flat plate thermal convection an increase in the velocity but a decrease in the
boundary layer flow of a Walters-B fluid using temperature and concentration. Prasad et al. [38]
numerical shooting quadrature. Chang et al [27] studied Radiation effects on MHD unsteady free
analyzed the unsteady buoyancy-driven flow and convection flow with mass transfer past a vertical
species diffusion in a Walters-B viscoelastic flow plate with variable surface temperature and
along a vertical plate with transpiration effects. concentration Owing to the significance of this
They showed that the flow is accelerated with a rise problem in chemical and medical biotechnological
in viscoelasticity parameter with both time and processing (e.g. medical cream manufacture).
distances close to the plate surface and that Therefore the objective of the present paper is to
increasing Schmidt number suppresses both velocity investigate the effect of thermophoresis on an
and concentration in time whereas increasing unsteady free convective heat and mass transfer
species Grashof number (buoyancy parameter) flow past a semi infinite vertical plate using the
accelerates flow through time. Hydrodynamic robust Walters-B viscoelastic rheologicctal material
stability studies of Walters-B viscoelastic fluids model.A Crank-Nicolson finite difference scheme is
were communicated by Sharma and Rana [28] for utilized to solve the unsteady dimensionless,
the rotating porous media suspension regime and by transformed velocity, thermal and concentration
Sharma et al [29] for Rayleigh-Taylor flow in a boundary layer equations in the vicinity of the
porous medium. Chaudhary and Jain [30] studied vertical plate. The present problem has to the
the Hall current and cross-flow effects on free and author’ knowledge not appeared thus far in the
forced Walters-B viscoelastic convection flow with literature. Another motivation of the study is to er
thermal radiative flux effects. Mahapatra et al [31] observed high heat transfer performance commonly
examined the steady two-dimensional stagnation- attributed to extensional investigate thestresses in
point flow of a Walters-B fluid along a flat viscoelastic boundary layers [25]

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2. CONSTITUTIVE EQUATIONS FOR This rheological model is very versatile and robust
THE WALTERS-B VISCOELASTIC and provides a relatively simple mathematical
FLUID formulation which is easily incorporated into
Walters [23] has developed a physically boundary layer theory for engineering applications
accurate and mathematically amenable model for [25, 26].
the rheological equation of state of a viscoelastic
fluid of short memory. This model has been shown 3. MATHEMATICAL MODEL:
to capture the characteristics of actual viscoelastic An unsteady two-dimensional laminar free
polymer solutions, hydrocarbons, paints and other convective flow of a viscoelastic fluid past a semi-
chemical engineering fluids. The Walters-B model infinite vertical plate is considered. The x-axis is
generates highly non-linear flow equations which taken along the plate in the upward direction and the
are an order higher than the classical Navier-Stokes y-axis is taken normal to it. The physical model is
(Newtonian) equations. It also incorporates elastic shown in Fig.1a.
properties of the fluid which are important in
extensional behavior of polymers. The constitute u
equations for a Walters-B liquid in tensorial form v
may be presented as follows: X
pik   pgik  pik * (1) Boundary
1 Tw , Cw layer
pik *  2    t  t * e1ik  t *  dt * (2)
T , C


N  
 t  t*   


e   t  t *  /  d  (3) o Y
0

where pik is the stress tensor, p is arbitrary isotropic pressure,
g ik is the metric tensor of a fixed coordinate system xi, eik 1 is
the rate of strain tensor and N   is the distribution function
of relaxation times, . The following generalized form of (2) has Fig.1a. Flow configuration and coordinate system
been shown by Walters [23] to be valid for all classes of motion
and stress.
Initially, it is assumed that the plate and the
x
1
pik  x, t   2    t  t  *m *r e1mr  x*t *  dt * (4)
* * fluid are at the same temperature T and 

x x 
concentration level C everywhere in the fluid. At
time, t  >0, Also, the temperature of the plate and
in which xi  xi
* *
 x, t, t  denotes the position at
*
the concentration level near the plate are raised to
time t* of the element which is instantaneously at  
Tw and C w respectively and are maintained
the position, xi, at time, t. Liquids obeying the
constantly thereafter. It is assumed that the
relations (1) and (4) are of the Walters-B’ type. For
such fluids with short memory i.e. low relaxation concentration C  of the diffusing species in the
times, equation (4) may be simplified to: binary mixture is very less in comparison to the
other chemical species, which are present, and hence
the Soret and Dufour effects are negligible. It is also
e1ik
p*ik  x, t   20e   2k0 (5) assumed that there is no chemical reaction between
1 ik

t the diffusing species and the fluid. Then, under the
 above assumptions, the governing boundary layer
in which 0   N  d defines the limiting equations with Boussinesq’s approximation are
0
viscosity at low shear rates, u v
Walters-B’  0 (6)

x y
k0    N  d is the Walters-B’ viscoelasticity
0


parameter and is the convected time derivative.
t

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u u u k Tw  T 
u v  g  T   T   g  *  C   C  
  
t  x y T0
 2u  3u (7) Typical values of  are 0.01, 0.05 and 0.1
 2  k0 2 corresponding to approximate values of
y y t 
k Tw  T  equal to 3.15 and 30 k for a reference
T  T  T  k  2T 
u v  (8) temperature of T0 =300 k.
t  x y  c p y 2 On introducing the following non-dimensional
C  C  C   2C   quantities
u v  D 2   cvt  (9)
t  x y y y x yGr1/4 uLGr 1/2
X ,Y  ,U  ,
L L 
The initial and boundary conditions are
T   T
 C   C
 tL2 1/2 (11)
t   0 : u  0, v  0, T   T , C   C 0
  T ,C  , t  Gr ,

Tw  T  
Cw  C  
t   0 : u  0, v  0, T   Tw , C   Cw at y  0
 
 * (Cw  C )
  k Gr1/2
u  0, T   T , C   C at x  0
  N ,  0 2 ,
 (Tw  T )
  L
u  0, T   T  , C   C  as y (10)
   g  (Tw  T )
 
Gr  ,
Where u, v are velocity components in x u0 3
and y directions respectively, t  - the time, g – the   kL Tw  T 
 
Pr  , Sc  , 
acceleration due to gravity,  - the volumetric  D Tw
coefficient of thermal expansion,  - the
*
Equations (6), (7), (8), (9) and (10) are reduced to
volumetric coefficient of expansion with the following non-dimensional form
concentration, T  -the temperature of the fluid in U U
 0 (12)
the boundary layer, C  -the species concentration X Y

in the boundary layer, Tw - the wall temperature, U U U  2U  3U (13)
U V   T  NC   2

T - the free stream temperature far away from the t X Y Y 2 Y t
 
plate, C w - the concentration at the plate, C - the
free stream concentration in fluid far away from the
plate,  - the kinematic viscosity,  - the thermal T T T 1  2T
diffusivity,  - the density of the fluid and D - the U V  (14)
species diffusion coefficient.
t X Y Pr Y 2
In the equation (4), the thermophoretic velocity C C C 1  2C
U V  
Vt was given by Talbot et al. [39] as t X Y Sc Y 2
T kv T   C T  2T 
VT   kv   C 2  (15)
Tw y 1 
Tw Gr 4  Y Y Y 
Where Tw is some reference temperature, the value The corresponding initial and boundary conditions
of kv represents the thermophoretic diffusivity, and are
k is the thermophoretic coefficient which ranges in t  0 : U  0, V  0, T  0, C  0
value from 0.2 to 1.2 as indicated by Batchelor and t  0 : U  0, V  0, T  1, C  1 at Y  0,
Shen [40] and is defined from the theory of Talbot
et al [39] by U  0, V  0, C  0 as X  0
(16)
2Cs (g /  p  Ct K n ) 1  K n (C1  C2e  Cs / Kn ) 
  U  0, V  0, C  0 as Y  
k
(1  3Cm K n ) 1  2g /  p  2Ct K n  Where Gr is the thermal Grashof number,
Pr is the fluid Prandtl number, Sc is the Schmidt
A thermophoretic parameter  can be defined (see number, N is the buoyancy ratio parameter,  is the
Mills et al 2 and Tsai [41]) as follows; viscoelastic parameter and  is the thermophoretic
parameter.
To obtain an estimate of flow dynamics at the
barrier boundary, we also define several important

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rate functions at Y = 0. These are the dimensionless 1
wall shear stress function, i.e. local skin friction U in, 1  U in1j 1  U in, 1  U in1j 
1

4x
j 1, j 1,
function, the local Nusselt number (dimensionless
temperature gradient) and the local Sherwood U in, j 1  U in1, j 1  U in, j  U in1, j (18)
number (dimensionless species, i.e. contaminant
transfer gradient) are computed with the following Vi ,nj1  Vi ,nj11  Vi ,nj  Vi ,nj 1
mathematical expressions [48]  0
2 y
1  T 
 XGr 4   U in, 1  U in, j U in, 1  U in1j  U in, j  U in1, j
3  U   Y Y 0
 X  Gr 4  , Nu x 
j
U n j 1,

 , t
i, j
2 x
 Y Y 0 TY 0
1  C  U in, 1  U in, 1  U in, j 1  U in, j 1
1 1

 XGr 4   Vi ,nj j j

 Y Y 0 (17) 4y
ShX 
CY 0 U in, 1  2U in, 1  U in, 1  U in, j 1  2U in, j  U in, j 1
1 1
j j j

We note that the dimensionless model
2  y  (19)
2
defined by Equations (12) to (15) under conditions
(16) reduces to Newtonian flow in the case of U in, 1  2U in, 1  U in, 1  U in, j 1  2U in, j  U in, j 1
1 1
vanishing viscoelasticity i.e. when  = 0  j j j

2  y  t
2

4.NUMERICAL SOLUTION
In order to solve these unsteady, non-linear 1 Ti ,nj1  Ti ,nj Cin, 1  Cin, j
N
4 j
coupled equations (12) to (15) under the conditions
Gr
2 2
(16), an implicit finite difference scheme of Crank-
Nicolson type has been employed. This method was
originally developed for heat conduction problems
Ti ,nj1  Ti ,nj Ti ,nj1  Ti 1,1j  Ti ,nj  Ti 1, j
n n
[42]. It has been extensively developed and remains  U in, j 
one of the most reliable procedures for solving t 2 x
partial differential equation systems. It is
unconditionally stable. It utilizes a central Ti ,nj11  Ti ,nj11  Ti ,nj 1  Ti ,nj 1 (20)
differencing procedure for space and is an implicit V n

4y
i, j
method. The partial differential terms are converted
n 1 n 1 n 1
1 Ti , j 1  2Ti , j  Ti , j 1  Ti , j 1  2Ti , j  Ti , j 1
to difference equations and the resulting algebraic n n n

problem is solved using a triadiagonal matrix
2  y 
2
algorithm. For transient problems a trapezoidal rule Pr
is utilized and provides second-order convergence.
The Crank-Nicolson Method (CNM) scheme has
Cin, 1  Cin, j Cin, 1  Cin1j  Cin, j  Cin1, j
been applied to a rich spectrum of complex j
U n j 1,

t 2 x
multiphysical flows. Kafousias and Daskalakis [43] i, j

have employed the CNM to analyze the
hydromagnetic natural convection Stokes flow for Cin, 1  Cin, 1  Cin, j 1  Cin, j 1
1 1


n j j
air and water. Edirisinghe [44] has studied V
4y
i, j
efficiently the heat transfer in solidification of
n 1 n 1 n 1
1 Ci , j 1  2Ci , j  Ci , j 1  Ci , j 1  2Ci , j  Ci , j 1
ceramic-polymer injection moulds with CNFDM. n n n

Sayed-Ahmed [45] has analyzed the laminar 
2  y 
2
dissipative non-Newtonian heat transfer in the Sc
entrance region of a square duct using CNDFM.
Nassab [46] has obtained CNFDM solutions for the   Cin, 1  Cin, 1  Cin, j 1  Cin, j 1 
j
1
j
1
(21)
unsteady gas convection flow in a porous medium 1   

with thermal radiation effects using the Schuster- Gr 4  4y 
Schwartzchild two-flux model. Prasad et al [47]  Ti , j 1  Ti , j 1  Ti , j 1  Ti , j 1 
n 1 n 1 n n
studied the combined transient heat and mass 
 

transfer from a vertical plate with thermal radiation
 4y 
effects using the CNM method. The CNM method
n 1 n 1 n 1
works well with boundary-layer flows. The finite  n
Ti , j 1  2Ti , j  Ti , j 1  Ti ,nj 1  2Ti ,nj  Ti ,nj 1
difference equations corresponding to equations (12) Ci , j
2  y 
1 2
to (15) are discretized using CNM as follows Gr 4

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Here the region of integration is considered as a 5.RESULTS AND DISCUSSION
rectangle with X max  1 , Ymax  14 and where In order to get a physical insight into the
problem, a parametric study is carried out to
Ymax corresponds to Ywhich lies well outside illustrate the effect of various governing
both the momentum and thermal boundary layers. thermophysical parameters on the velocity,
The maximum of Y was chosen as 14, after some temparature, concentration, skin-friction, Nusselt
preliminary numerical experiments such that the last number and Sherwood number are shown in figures
two boundary conditions of (19) were satisfied and tables.
5 In figures 2(a) to 2(c) we have presented
within the tolerance limit 10 . The mesh sizes have
been fixed the variation of velocity U, temparature T and
as XYwith
 concentration C versus (Y) with collective effects of
time
step  t0.01. The computations are executed thermophoretic parameter () at X = 0 for opposing
initially by reducing the spatial mesh sizes by 50% flow (N<0). In case of Newtonian fluids ( = 0), an
in one direction, and later in both directions by 50%. increase in  from 0.0 through 0.5 to maximum
The results are compared. It is observed that, in all value of 1.0 as depicted in figure 2(a) for opposing
the cases, the results differ only in the fifth decimal flow (N < 0) . Clearly enhances the velocity U
place. Hence, the choice of the mesh sizes is verified which ascends sharply and peaks in close vicinity to
as extremely efficient. The coefficients of the plate surface (Y=0). With increasing distance
from the plate wall however the velocity U is
Uik, j andVi,kj ,appearing in the finite difference adversely affected by increasing thermophoretic
equations are treated as constant at any one-time effect i.e. the flow is decelerated. Therefore close to
step. Here i designates the grid point along the X- the plate surface the flow velocity is maximized for
direction, j along the Y-direction and k in the time the case ofBut this trend is reversed as
variable, t. The values of U, V, T and C are known at we progress further into the boundary layer regime.
all grid points when t = 0 from the initial conditions. The switchover in behavior corresponds to
The computations for U, V, T and C at a time level approximately Y =3.5, with increasing velocity
(k + 1), using the values at previous time level k are profiles decay smoothly to zero in the free stream at
carried out as follows. The finite-difference equation the edge of the boundary layer. The opposite effect
(21) at every internal nodal point on a particular is caused by an increase in time.A rise in  from
ilevel constitutes a tri diagonal system of
- 6.36, 7.73 to 10.00 causes a decrease in flow
equations and is solved by Thomas algorithm as velocity U near the wall in this case the maximum
discussed in Carnahan et al. [45]. Thus, the values velocity arises for the least time progressed.With
of C are known at every nodal point at a particular i more passage of time t = 10.00 the flow is
at  k  1 time level. Similarly, the values of U
th decelerated.Again there is a reverse in the response
at Y =3.5, and thereafter velocity is maximized with
and T are calculated from equations (19), (20) the greatest value of time. A similar response is
respectively, and finally the values of V are observed for the non-Newtonian fluid (   0 ), but
calculated explicitly by using equation (18) at every clearly enhances the velocity very sharply and peaks
nodal point on a particular i level at  k  1
th highly in close vicinity to the plate surface
compared in case of Newtonian fluid.
time level. In a similar manner, computations are In figure 2(b), in case of Newtonian fluids
carried out by moving along i -direction. After
(=0) and non-Newtonian fluids (   0 ), the
computing values corresponding to each i at a time
thermophoretic parameter is seen to increase
level, the values at the next time level are
temperature throughout the bounder layer.All
determined in a similar manner. Computations are
profiles increases from the maximum at the wall to
repeated until steady state is reached. The steady
zero in the free stream. The graphs show therefore
state solution is assumed to have been reached when
that increasing thermophoretic parameter heated
the absolute difference between the values of the
the flow. With progression of time, however the
velocity U, temperature T, as well as concentration
5
temperature T is consistently enhanced i.e. the fluid
C at two consecutive time steps are less than 10 at is cool as time progress.
all grid points. The scheme is unconditionally stable.
The local truncation error is
O(t 2  X 2  Y 2 ) and it tends to zero as t,
X, and  Ytend to zero. It follows that the
CNM scheme is compatible. Stability and
compatibility ensure the convergence.

2085 | P a g e


In figure 2(c) theopposite response is
observed for the concentration field C. In case of
The switchover in behavior corresponds to
Newtonian fluids (=0) and non-Newtonian fluids (
approximately Y =3.5, with increasing velocity
  0 ), the thermophoretic parameter  profiles decay smoothly to zero in the free stream at
increases, the concentration throughout the the edge of the boundary layer.The opposite effect is
boundary layer regime (0<Y<14) decreased. caused by an increase in time. A rise in time t from
6.36, 7.73 to 10.00 causes a decrease in flow

In figures 3(a) to 3(c) we have presented
the variation of velocity U, temparature T and
concentration C versus (Y) with collective effects of
thermophoretic parameter at X = 0 for aiding velocity U near the wall in this case the
flow. In case of Newtonian fluids ( = 0), an maximum velocity arises for the least time
increase in  from 0.0 through 0.5 to maximum progressed.With more passage of time t = 10.00 the
value of 1.0 as depicted in figure 3(a) for aiding flow is decelerated. Again there is a reverse in the
flow (N>0). Clearly enhances the velocity U which response at Y =3.5, and thereafter velocity is
ascends sharply and peaks in close vicinity to the maximized with the greatest value of time. A similar
plate surface (Y=0). With increasing distance from response is observed for the non-Newtonian fluid (
the plate wall however the velocity U is adversely   0 ), but clearly enhances the velocity very
affected by increasing thermophoretic effect i.e. sharply and peaks highly in close vicinity to the
theflow is decelerated. Therefore close to the plate plate surface compared in case of Newtonian fluid.
surface the flow velocity is maximized for the case
of But this trend is reversed as we progress
further into the boundary layer regime.

2086 | P a g e

depicted in figure 4(a), clearly enhances the velocity
U which ascends sharply and peaks in close vicinity
to the plate surface (Y=0),with increasing distance
from the plate wall the velocity U is adversely
affected by increasing viscoelasticity i.e. the flow is
decelerated. Therefore close to the plate surface the
flow velocity is maximized for the case ofnon-
Newtonian fluid(   0 ). The switchover in
behavior corresponds to approximately Y=2.

In figure 3(b), in case of Newtonian fluids
(=0) and non-Newtonian fluids (   0 ), the
thermophoretic parameter  is seen to decrease
temperature throughout the bounder layer. All
profiles decreases from the maximum at the wall to
zero in the free stream. The graphs show therefore
that increasing thermophoretic parameter cools the With increasing Y, velocity profiles decay
flow. With increasing of time t, the temperature T is smoothly to zero in the free stream at the edge of
consistently enhanced i.e. the fluid is heated as time theboundary layer. Pr<1 physically corresponds to
progress. cases where heat diffuses faster than momentum. In
In figure 3(c) a similar response is the case of water based solvents i.e. Pr = 7.0, a
observed for the concentration field C. In case of similar response is observed for the velocity field in
Newtonian fluids (=0) and non-Newtonian fluids ( figure 4(a).
In figure 4(b), in case of air based solvents
  0 ), the thermophoretic parameter  increases, i.e. Pr = 0.71, an increase in viscoelasticity 
the concentration throughout the boundary layer increasing from 0.000, 0.003 to 0.005, temperature
regime (0<Y<14) decreased. All profiles decreases T is markedly reduced throughout the boundary
from the maximum at the wall to zero in the free layer. In case of water based solvents i.e. Pr = 7.0
stream. also a similar response is observed, but it is very
Figures 4(a) to 4(c) illustrate the effect of closed to the plate surface. The descent is
Prandtl number (Pr), Viscoelastic parameter () and increasingly sharper near the plate surface for higher
time t on velocity (U), temperature (T) and Pr values a more gradual monotonic decay is
concentration (C) without thermophoretic effect witnessed smaller Pr values in this case, cause a
(=0) at X=1.0.Pr defines the ratio of momentum thinner thermal boundary layer thickness and more
diffusivity () to thermal diffusivity. In case of air uniform temperature distributions across the
based solvents i.ePr = 0.71, an increase in  from boundary layer. Smaller Pr fluids possess higher
0.000, 0.003 and the maximum value of 0.005 as thermal conductivities so that heat can diffuse away

2087 | P a g e

from the plate surface faster than for higher Pr fluids T is markedly reduced throughout the boundary
(thicker boundary layers). Our computations show layer. In case of water based solvents i.e. Pr = 7.0
that a rise in Pr depresses the temperature function, also a similar response is observed, but it is very
a result constant with numerous other studies on closed to the plate surface. A similar response is
coupled heat and mass transfer. For the case of Pr = observed for the concentration field C in figure
1, thermal and velocity boundary layer thickness are 5(c).In both cases Pr = 0.71 and Pr = 7.0, when 
equal. increasing from 0.000, 0.003 to 0.005, concentration
A similar response is observed for the C also reduced throughout the boundary layer
concentration field C in figure 4(c). In both cases Pr regime (0<Y<14). All profiles decreases from the
= 0.71 and Pr = 7.0, when  increasing from 0.000, maximum at the wall to zero in the free stream.
0.003 to 0.005, concentration C also reduced
throughout the boundary layer regime (0<Y<14).
All profiles decreases from the maximum at the wall
to zero in the free stream.

Figures 6(a) to 6(c) depict the distributions
of velocity U, temperature T and concentration C
versus coordinate (Y) for various Schmidtnumbers
(Sc) with collective effects of thermophoretic
parameter () in case of Newtonian fluids (=0)
and time (t), close to the leading edge at X = 1.0, are
shown. Correspond to Schmidt number Sc=0.6 an
increase in from 0.0 through 0.5 to 1.0 as
depicted in figure 6(a), clearly enhances the velocity
U which ascends sharply and peaks in close vicinity
to the plate surface (Y=0). With increasing distance
from the plate wall however the velocity U is
adversely affected by increasing thermophoretic
effect i.e. the flow is decelerated.

Figures 5(a) to 5(c) illustrate the effect of
Prandtl number (Pr), Viscoelastic parameter () and
time t on velocity U, temperature T and
concentration C with thermophoretic effect ( =
0.5) at X=1.0. In case of air based solvents i.ePr =
0.71, an increase in  from 0.000, 0.003 and the
maximum value of 0.005 as depicted in figure 5(a),
clearly enhances the velocity U which ascends
sharply and peaks in close vicinity to the plate
surface (Y=0). With increasing Y, velocity profiles
decay smoothly to zero in the free stream at the edge Therefore close to the plate surface the
of the boundary layer. flow velocity is maximized for the case of
In figure 5(b), in case of air based solvents But this trend is reversed as we progress
i.e. Pr = 0.71, an increase in viscoelasticity  further into the boundary layer regime. The
increasing from 0.000, 0.003 to 0.005, temperature switchover in behavior corresponds to

2088 | P a g e

profiles decay smoothly to zero in the free stream at
the edge of the boundary layer.A similar response is
observed in case of Schmidt number Sc=2.0 also.

maximum concentrations will be associated
with this case (Sc = 0.6).For Sc > 1, momentum will
diffuse faster than species causing progressively
lower concentration values. With a increase in
molecular diffusivity concentration boundary layer
With higher Sc values the gradient of
thickness is therefore increased. For the special case
velocity profiles is lesser prior to the peak velocity
of Sc = 1, the species diffuses at the same rate as
but greater after the peak.
momentum in the viscoelastic fluid. Both
In figure 6(b), in case of Newtonian fluids
concentration and boundary layer thicknesses are
(=0) and for Schmidt number Sc=0.6, and 2.0 the
the same for this case. An increase in Schmidt
increasing in thermophoretic parameter is seen to
number effectively depresses concentration values
decrease the temperature throughout the bounder
in the boundary layer regime since higher Sc values
layer. All profiles decreases from the maximum at
will physically manifest in a decrease of molecular
the wall to zero in the free stream. The graphs show
diffusivity (D) of the viscoelastic fluid i.e. a
therefore that decreasing thermophoretic parameter
reduction in the rate of mass diffusion. Lower Sc
cools the flow. With progression of time, however
values will exert the reverse influence since they
the temperature T is consistently enhanced i.e. the
correspond to higher molecular diffusivities.
fluid is heated as time progress.
Concentration boundary layer thickness is therefore
In figure 6(c) a similar response is
considerably greater for Sc = 0.6 than for Sc = 2.0.
observed for the concentration field C. In case of
Figures 7(a) to 7(c) depict the distributions of
Newtonian fluids (=0) and for Schmidt number
velocity U, temperature T and concentration C
Sc=0.6, and 2.0, the increasing in thermophoretic
versus coordinate (Y) for various Schmidt numbers
parameter  increases the concentration
(Sc) with collective effects of thermophoretic
throughout the boundary layer regime (0<Y<14).
parameter () in case of non-Newtonian fluids(
All profiles increases from the maximum at the wall
to zero in the free stream. Sc defines the ratio of   0 ) and time (t), close to the leading edge at X
momentum diffusivity () to molecular diffusivity = 1.0, are shown. Correspond to Schmidt number
(D). For Sc<1, species will diffuse much faster than Sc=0.6 an
momentum so that

2089 | P a g e

In figure 7(c) a similar response is observed for the
concentration field C. In case of non-Newtonian
fluids (   0 ) and for Schmidt number Sc=0.6, and
2.0, the increasing in thermophoretic parameter 
increases the concentration throughout the boundary
layer regime (0<Y<14). All profiles increases from
the maximum at the wall to zero in the free stream.
Sc defines the ratio of momentum diffusivity (n) to
molecular diffusivity (D). For Sc<1, species will
diffuse much faster than momentum so that
maximum concentrations will be associated with
this case (Sc = 0.6). For Sc > 1, momentum will
diffuse faster than species causing progressively
lower concentration values. With a increase in
molecular diffusivity concentration boundary layer
thickness is therefore increased. For the special case
of Sc = 1, the species diffuses at the same rate as
momentum in the viscoelastic fluid. Both
concentration and boundary layer thicknesses are
the same for this case. An increase in Schmidt
number effectivelydepresses concentration values in
the boundary layer regime since higher Sc values
will physically manifest in a decrease of molecular
diffusivity (D) of the viscoelastic fluid i.e. a
reduction in the rate of mass diffusion. Lower Sc
values will exert the reverse influence since they
correspond to higher molecular diffusivities.
Concentration boundary layer thickness is therefore
considerably greater for Sc = 0.6 than for Sc = 2.0.
Figures 8a to 8c present the effects of buoyancy
increase in  from 0.0 through 0.5 to 1.0 ratio parameter, N on U, T and C profiles. The
as depicted in figure 7(a),clearly enhances the maximum time elapse to the steady state scenario
velocity U which ascends sharply and peaks in close accompanies the only negative value of N i.e. N = -
vicinity to the plate surface (Y=0). With increasing 0.5. For N = 0 and then increasingly positive values
distance from the plate wall however the velocity U of N up to 5.0, the time taken, t, is steadily reduced.
is adversely affected by increasing thermophoretic As such the presence of aidingbuoyancy forces
effect i.e. the flow is decelerated. Therefore close to (both thermal and species buoyancy force acting in
the plate surface the flow velocity is maximized for unison) serves to stabilize the transient flow regime.
the case of But this trend is reversed as we
 *  Cw  C 
 
progress further into the boundary layer regime. The The parameter N  and expresses
switchover in behavior corresponds to  Tw  T 
 
the ratio of the species (mass diffusion) buoyancy
profiles decay smoothly to zero in the free stream at
force to the thermal (heat diffusion) buoyancy force.
the edge of the boundary layer. A similar response is
When N = 0 the species buoyancy term, NC
observed in case of Schmidt number Sc=2.0 also.
vanishes and the momentum boundary layer
All profiles descend smoothly to zero in the free
equation (13) is de-coupled from the species
stream. With higher Sc values the gradient of
diffusion (concentration) boundary layer equation
velocity profiles is lesser prior to the peak velocity
(15). Thermal buoyancy does not vanish in the
but greater after the peak.
momentum equation (13) since the term T is not
In figure 7(b), in case of non-Newtonian
affected by the buoyancy ratio.
fluids (   0 ) and for Schmidt number Sc=0.6, and
2.0 the increasing in thermophoretic parameter is
seen to decrease the temperature throughout the
bounder layer. All profiles decreases from the
maximum at the wall to zero in the free stream. The
graphs show therefore that decreasing
thermophoretic parameter cools the flow. With
progression of time, however the temperature T is
consistently enhanced i.e. the fluid is heated as time
progress.

2090 | P a g e

In figures 9a to 9c the variation of dimensionless
local skin friction (surface shear stress),  X Nusselt
,
number (surface heat transfer gradient), Nu X and
the Sherwood number (surface concentration
gradient), ShX , versus axial coordinate (X) for
various viscoelasticity parameters () and time tare
illustrated.Shear stress is clearly enhanced with
increasing viscoelasticity (i.e. stronger elastic
effects) i.e. the flow is accelerated, a trend
consistent with our earlier computations in figure
9a.

The ascent in shear stress is very rapid
from the leading edge (X = 0) but more gradual as
we progress along the plate surface away from the
plane.With an increase in time, t, shear stress,  X is
When N < 0 we have the case of opposing
buoyancy. An increase in N from -0.5, through 0, 1, increased.Increasing viscoelasticity () is observed
2, 3, to 5 clearly accelerates the flow i.e. induces a in figure 9b to enhance local Nusselt number, Nu X
strong escalation in stream wise velocity, U, close to values whereas they areagain increased with greater
the wall; thereafter velocities decay to zero in the time.Similarly in figure 9c the local Sherwood
free stream. At some distance from the plate surface,
approximately Y = 2.0, there is a cross-over in number ShX
profiles. Prior to this location the above trends are
apparent. However after this point, increasingly
positive N values in fact decelerate the flow.
Therefore further from the plate surface, negative N
i.e. opposing buoyancy is beneficial to the flow
regime whereas closer to the plate surface it has a
retarding effect. A much more consistent response
to a change in the N parameter is observed in figure
8b, where with a rise from -0.5 through 0, 1.0, 2.0,
3.0 to 5.0 (very strong aiding buoyancy case) the
temperature throughout the boundary layer is
strongly reduced. As with the velocity field (figure
8a), the time required to attain the steady state
decreases substantially with a positive increase in N.
Aiding (assisting) buoyancy therefore stabilizes the
temperature distribution.A similar response is
evident for the concentration distribution C, Whichs
shown in figure 8c, also decreases with
positiveincrease in N but reaches the steady state
progressively faster.

2091 | P a g e

increase in time (t) both Shear stress and local
Nusselt number are enhanced.

values are elevated with an increase in elastic effects
i.e. a rise in from 0 (Newtonian flow) though r
0.001, 0.003, 0.005 to 0.007 but depressed slightly
with time.

Finally in figures 10a to 10c the influence of
Thermophoretic parameter and time (t) on  X With increasing  values, local Sherwood number,
,
Nu X and ShX , versus axial coordinate (X) are ShX , as shown in figure 10c, is boosted
depicted. considerably along the plate surface;gradients of the
profiles are also found to diverge with increasing X
values. However an increase in time, t, serves to
increase local Sherwood numbers.

6. CONCLUSIONS
A two-dimensional, unsteady laminar
incompressible boundary layer model has been
presented for the external flow, heat and mass
transfer in a viscoelastic buoyancy-driven flow past
a semi-infinite vertical plate under the influence of
thermophoresis. The Walters-B viscoelastic model
has been employed which is valid for short memory
polymeric fluids. The dimensionless conservation
equations have been solved with the well-tested,
robust, highly efficient, implicit Crank Nicolson
finite difference numerical method. The present
computations have shown that increasing
An increase in from 0.0 through 0.3, 0.5, 1.0 to viscoelasticity accelerates the velocity and enhances
1.5, strongly increases both  X and Nu X along the shear stress (local skin friction), local Nusselt
entire plate surface i.e. for all X. However with an number and local Sherwood number, but reduces

2092 | P a g e

temperature and concentration in the boundary aerosol- particle deposition from a laminar
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